Sauté onion in butter; set aside.
In a medium saucepan, over medium heat, cook melted butter and flour, using a whisk, for 3 - 5 minutes.
Stir constantly and add the half and half.
Add the chicken stock; simmer for 20 minutes.
Add the broccoli, carrots and onions.
Cook, over low heat, for 20 - 25 minutes.
Add salt and pepper, to taste.
Can be puréed in a blender but serving without blending is also very good.
After blending, return to heat and add cheese.
Stir in nutmeg.